The Church numbers in NF set theory

TL;DR

This paper defines Church numerals within INF set theory, proves their set is infinite using a new counting axiom, and shows that Heyting arithmetic can be interpreted in this framework.

Contribution

It introduces the Church counting axiom in INF set theory and proves the infinitude of Church numbers within an intuitionistic logic setting.

Findings

The set of Church numbers is infinite under the counting axiom.

Church successor is one-to-one without the counting axiom.

Heyting's arithmetic is interpretable in INF plus the Church counting axiom.

Abstract

By INF we mean Quine's NF set theory, with intuitionistic logic. We define the Church numerals (or better, Church numbers) and elaborate their properties in INF. The Church counting axiom says that iterating successor $n$ times, starting at zero, results in $n$. With the aid of the counting axiom we prove that the set of Church numbers is infinite. This is a new result even with classical logic; that is, just because there is some infinite set, it is not immediate that the set of Church numbers is infinite. Specker showed in 1953 that classical NF proves the existence of an infinite set. It has long been an open problem whether INF can prove that. Now we show that it can be done intuitionistically, with the aid of the Church counting axiom. We also prove, without the aid of the counting axiom, that if the set of Church numbers is not finite, then it is infinite, and Church successor is one-to-one. Consequently, Heyting's arithmetic is interpretable in INF plus the Church counting axiom.